Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

TL;DR

This paper proves that strongly summable ultrafilters on abelian groups are sparse and have the trivial sums property, and explores their relationship with union ultrafilters, including a construction under Martin's Axiom.

Contribution

It establishes that strongly summable ultrafilters are sparse and possess the trivial sums property, and shows their isomorphism to union ultrafilters in most cases, with a counterexample under Martin's Axiom.

Findings

Strongly summable ultrafilters are sparse.

Such ultrafilters have the trivial sums property.

Counterexample of a non-isomorphic ultrafilter under Martin's Axiom.

Abstract

We answer two questions of Hindman, Stepr\=ans and Strauss, namely we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases: we also construct (assuming Martin's Axiom for countable partial orders, i.e. $\mathrm{cov}(\mathcal M)=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter.